What is the ruling on delaying Asr prayer on Friday, or the remaining prayers, until, for example, an hour before Maghrib, with the intention of making the recommended supplication at that time?

The Sunnah (definition) is to pray Asr early, performing it at the beginning of its time without exception. This is supported by the guidance of the Prophet, peace and blessings be upon him, and the sayings of the Companions and the early generations (Salaf).

Supplication during the hour of response on Friday or at the time of rain does not require it to be during prayer. Rather, supplication is answered, Allah willing, for whoever encounters these times, even if they are not praying. Moreover, one who sits awaiting prayer is considered to be in prayer.